POINTS SURROUNDING THE ORIGIN

Abstract

Suppose d > 2, n > d+ 1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p is an element of P, the convex hull of QU{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A(1),...,A(d+1) in R(d) if the origin is contained in the convex hull of A(i)UA(j) for all 1 <= i < j <= d+1, then there is a simplex S containing the origin such that vertical bar S boolean AND A(i)vertical bar=1 for every 1 < i <= d+1. This is a generalization of Barany's colored Caratheodory theorem, and in a dual version, it gives a spherical version of Lovasz' colored Helly theorem.